4,446 research outputs found
A CMV--based eigensolver for companion matrices
In this paper we present a novel matrix method for polynomial rootfinding. By
exploiting the properties of the QR eigenvalue algorithm applied to a suitable
CMV-like form of a companion matrix we design a fast and computationally simple
structured QR iteration.Comment: 14 pages, 4 figure
Fast Hessenberg reduction of some rank structured matrices
We develop two fast algorithms for Hessenberg reduction of a structured
matrix where is a real or unitary diagonal
matrix and . The proposed algorithm for the
real case exploits a two--stage approach by first reducing the matrix to a
generalized Hessenberg form and then completing the reduction by annihilation
of the unwanted sub-diagonals. It is shown that the novel method requires
arithmetic operations and it is significantly faster than other
reduction algorithms for rank structured matrices. The method is then extended
to the unitary plus low rank case by using a block analogue of the CMV form of
unitary matrices. It is shown that a block Lanczos-type procedure for the block
tridiagonalization of induces a structured reduction on in a block
staircase CMV--type shape. Then, we present a numerically stable method for
performing this reduction using unitary transformations and we show how to
generalize the sub-diagonal elimination to this shape, while still being able
to provide a condensed representation for the reduced matrix. In this way the
complexity still remains linear in and, moreover, the resulting algorithm
can be adapted to deal efficiently with block companion matrices.Comment: 25 page
An efficient multi-core implementation of a novel HSS-structured multifrontal solver using randomized sampling
We present a sparse linear system solver that is based on a multifrontal
variant of Gaussian elimination, and exploits low-rank approximation of the
resulting dense frontal matrices. We use hierarchically semiseparable (HSS)
matrices, which have low-rank off-diagonal blocks, to approximate the frontal
matrices. For HSS matrix construction, a randomized sampling algorithm is used
together with interpolative decompositions. The combination of the randomized
compression with a fast ULV HSS factorization leads to a solver with lower
computational complexity than the standard multifrontal method for many
applications, resulting in speedups up to 7 fold for problems in our test
suite. The implementation targets many-core systems by using task parallelism
with dynamic runtime scheduling. Numerical experiments show performance
improvements over state-of-the-art sparse direct solvers. The implementation
achieves high performance and good scalability on a range of modern shared
memory parallel systems, including the Intel Xeon Phi (MIC). The code is part
of a software package called STRUMPACK -- STRUctured Matrices PACKage, which
also has a distributed memory component for dense rank-structured matrices
A numerical comparison of solvers for large-scale, continuous-time algebraic Riccati equations and LQR problems
In this paper, we discuss numerical methods for solving large-scale
continuous-time algebraic Riccati equations. These methods have been the focus
of intensive research in recent years, and significant progress has been made
in both the theoretical understanding and efficient implementation of various
competing algorithms. There are several goals of this manuscript: first, to
gather in one place an overview of different approaches for solving large-scale
Riccati equations, and to point to the recent advances in each of them. Second,
to analyze and compare the main computational ingredients of these algorithms,
to detect their strong points and their potential bottlenecks. And finally, to
compare the effective implementations of all methods on a set of relevant
benchmark examples, giving an indication of their relative performance
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